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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution">Laplace Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">laplace</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">laplace_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">laplace</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">laplace_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
   <span class="comment">// Construct:</span>
   <span class="identifier">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          Laplace distribution is the distribution of differences between two independent
          variates with identical exponential distributions (Abramowitz and Stegun
          1972, p. 930). It is also called the double exponential distribution.
        </p>
<p>
          For location parameter <span class="emphasis"><em>μ</em></span> and scale parameter <span class="emphasis"><em>σ</em></span>,
          it is defined by the probability density function:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span>

          </p></blockquote></div>
<p>
          The location and scale parameters are equivalent to the mean and standard
          deviation of the normal or Gaussian distribution.
        </p>
<p>
          The following graph illustrates the effect of the parameters μ and σ on the
          PDF. Note that the domain of the random variable remains [-∞,+∞] irrespective
          of the value of the location parameter:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.member_functions"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span>
          and scale <span class="emphasis"><em>scale</em></span>.
        </p>
<p>
          The location parameter is the same as the mean of the random variate.
        </p>
<p>
          The scale parameter is proportional to the standard deviation of the random
          variate.
        </p>
<p>
          Requires that the scale parameter is greater than zero, otherwise calls
          <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is [-∞,+∞].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.accuracy"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The laplace distribution is implemented in terms of the standard library
          log and exp functions and as such should have very small errors.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.implementation"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table μ is the location parameter of the distribution, σ is
          its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
          is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = e<sup>-abs(x-μ) / σ</sup> / (2 * σ)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relations:
                  </p>
                  <p>
                    x &lt; μ : p = e<sup>(x-μ)/σ </sup> / σ
                  </p>
                  <p>
                    x &gt;= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ

                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation:
                  </p>
                  <p>
                    -x &lt; μ : q = e<sup>(-x-μ)/σ </sup> / σ
                  </p>
                  <p>
                    -x &gt;= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ

                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relations:
                  </p>
                  <p>
                    p &lt; 0.5 : x = μ + σ * log(2*p)
                  </p>
                  <p>
                    p &gt;= 0.5 : x = μ - σ * log(2-2*p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation:
                  </p>
                  <p>
                    q &gt; 0.5: x = μ + σ*log(2-2*q)
                  </p>
                  <p>
                    q &lt;=0.5: x = μ - σ*log( 2*q )
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    μ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    2 * σ<sup>2</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    μ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    6
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    3
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.references"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.references">References</a>
        </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein,
              Eric W. "Laplace Distribution."</a> From MathWorld--A
              Wolfram Web Resource.
            </li>
<li class="listitem">
              <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace
              Distribution</a>
            </li>
<li class="listitem">
              M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
              1972, p. 930.
            </li>
</ul></div>
</div>
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      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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